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\BOOKMARK [1][-]{section.1}{Introduction}{}% 1
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\BOOKMARK [1][-]{section.2}{Introduction to Neural Networks}{}% 2
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\BOOKMARK [2][-]{subsection.2.1}{Nonlinearity of Neural Networks}{section.2}% 3
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\BOOKMARK [2][-]{subsection.2.2}{Training Neural Networks}{section.2}% 4
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\BOOKMARK [3][-]{subsubsection.2.2.1}{Nonlinearity in the Last Layer}{subsection.2.2}% 5
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\BOOKMARK [3][-]{subsubsection.2.2.2}{Error Measurement}{subsection.2.2}% 6
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\BOOKMARK [3][-]{subsubsection.2.2.3}{Gradient Descent Algorithm}{subsection.2.2}% 7
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\BOOKMARK [1][-]{section.3}{Shallow Neural Networks}{}% 8
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\BOOKMARK [2][-]{subsection.3.1}{Convergence Behavior of One-Dimensional Randomized Shallow Neural Networks}{section.3}% 9
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\BOOKMARK [2][-]{subsection.3.2}{Simulations}{section.3}% 10
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\BOOKMARK [1][-]{section.4}{Application of Neural Networks to Higher Complexity Problems}{}% 11
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\BOOKMARK [2][-]{subsection.4.1}{Convolution}{section.4}% 12
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\BOOKMARK [2][-]{subsection.4.2}{Convolutional Neural Networks}{section.4}% 13
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\BOOKMARK [2][-]{subsection.4.3}{Stochastic Training Algorithms}{section.4}% 14
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\BOOKMARK [2][-]{subsection.4.4}{Modified Stochastic Gradient Descent}{section.4}% 15
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\BOOKMARK [2][-]{subsection.4.5}{Combating Overfitting}{section.4}% 16
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\BOOKMARK [3][-]{subsubsection.4.5.1}{Dropout}{subsection.4.5}% 17
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\BOOKMARK [3][-]{subsubsection.4.5.2}{Manipulation of Input Data}{subsection.4.5}% 18
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\BOOKMARK [3][-]{subsubsection.4.5.3}{Comparisons}{subsection.4.5}% 19
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\BOOKMARK [3][-]{subsubsection.4.5.4}{Effectiveness for Small Training Sets}{subsection.4.5}% 20
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\BOOKMARK [1][-]{section.5}{Summary and Outlook}{}% 21
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\BOOKMARK [1][-]{section*.27}{Appendices}{}% 22
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\BOOKMARK [1][-]{Appendix.1.A}{Notes on Proofs of Lemmata in Section 3.1}{}% 23
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\BOOKMARK [1][-]{Appendix.1.B}{Implementations}{}% 24
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\BOOKMARK [1][-]{Appendix.1.C}{Additional Comparisons}{}% 25
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